Apply the Properties of Integer Exponents to generate equivalent expressions to 37⋅3−9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
3^7⋅3^−9=
To apply the property of integer exponents, we can combine the exponents when multiplying the same base. The property states that when multiplying powers with the same base, add the exponents.
Using this property, we can rewrite the expression as:
3^7⋅3^(-9) = 3^(7 + (-9))
Now, we can simplify the exponent:
3^(7 + (-9)) = 3^(-2)
Since the exponent is negative, we can rewrite it as a positive exponent by moving the base to the denominator:
3^(-2) = 1/3^2
Simplifying further, we have:
1/3^2 = 1/9
Therefore, the equivalent expression to 37⋅3^(-9) with only positive exponents is 1/9.
Apply the Properties of Integer Exponents to generate equivalent expressions to 2^2/2^−4 with only positive exponents, then solve the expression. Provide the solution as a whole number.(1 point)
2^2/2^−4=
To apply the property of integer exponents, we can subtract the exponents when dividing the same base. The property states that when dividing powers with the same base, subtract the exponents.
Using this property, we can rewrite the expression as:
2^2/2^(-4) = 2^(2 - (-4))
Now, we can simplify the exponent:
2^(2 - (-4)) = 2^(2 + 4)
2^(2 + 4) = 2^6
Therefore, the equivalent expression to 2^2/2^(-4) with only positive exponents is 2^6.
Simplifying further, we have:
2^6 = 64
Therefore, 2^2/2^(-4) is equal to 64.
Apply the Properties of Integer Exponents to generate equivalent expressions to 4^0 ⋅ 5^−2/(−2)^−3 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
4^0 ⋅ 5^−2/(−2)^−3=
To apply the properties of integer exponents, let's simplify each component of the expression individually.
First, let's simplify 4^0. Any number raised to the power of 0 is equal to 1. So, 4^0 = 1.
Next, let's simplify 5^(-2). Using the property of integer exponents, a negative exponent indicates that the base should be moved to the denominator with a positive exponent. So, 5^(-2) = 1/5^2 = 1/25.
Lastly, let's simplify (-2)^(-3). Again, using the property of integer exponents, a negative exponent indicates that the base should be moved to the denominator with a positive exponent. So, (-2)^(-3) = 1/(-2)^3 = 1/(-8) = -1/8.
Now, let's substitute the simplified values back into the original expression:
4^0 ⋅ 5^(-2)/(−2)^(-3) = 1 ⋅ 1/25 / (-1/8) = 1/25 / (-1/8)
Following the rules of division with fractions, we can multiply the numerator by the reciprocal of the denominator:
1/25 / (-1/8) = 1/25 * (-8/1) = -8/25
Therefore, the simplified fraction for the expression 4^0 ⋅ 5^(-2)/(−2)^(-3) with only positive exponents is -8/25.
Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3⋅27−7)^0/3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(14^3⋅27−7)^0/3^4=
To apply the properties of integer exponents, let's simplify each component of the expression individually.
First, let's simplify (14^3⋅27^(-7))^0. Any number raised to the power of 0 is equal to 1. So, (14^3⋅27^(-7))^0 = 1.
Next, let's simplify 3^4. 3^4 is equal to 3 * 3 * 3 * 3, which equals 81.
Now, let's substitute the simplified values back into the original expression:
(14^3⋅27^(-7))^0/3^4 = 1/81
Therefore, the simplified fraction for the expression (14^3⋅27^(-7))^0/3^4 with only positive exponents is 1/81.
Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
(4^3 ⋅ 7^−2)−5/7^9 ⋅ 4^−13=
To apply the properties of integer exponents, let's simplify each component of the expression individually.
First, let's simplify (43 ⋅ 7^(-2))^-5. To simplify the exponent, we can flip the base and make the exponent positive. So, (43 ⋅ 7^(-2))^-5 is equal to (1/(43 ⋅ 7^(-2)))^5.
Next, let's simplify 7^9. 7^9 is already in its simplest form.
Now, let's simplify 4^(-13). Similar to before, we can flip the base and make the exponent positive. So, 4^(-13) is equal to 1/(4^13).
Now, let's substitute the simplified values back into the original expression:
(4^3 ⋅ 7^(-2))^-5/7^9 ⋅ 4^(-13) = (1/(43 ⋅ 7^(-2)))^5/7^9 ⋅ 1/(4^13)
Moving the negative exponent to the denominator for 4^(-13), we get:
(1/(43 ⋅ 7^(-2)))^5/7^9 ⋅ 1/(1/4^13) = (1/(43 ⋅ 7^(-2)))^5/7^9 ⋅ 4^13
Applying the property of exponentiation with negative exponents for 7^(-2), we get:
(1/(43 ⋅ 1/7^2))^5/7^9 ⋅ 4^13 = (1/((43/1) ⋅ (1/49)))^5/7^9 ⋅ 4^13
Simplifying further:
(1/(43/49))^5/7^9 ⋅ 4^13 = (49/43)^5/7^9 ⋅ 4^13
Therefore, the simplified fraction for the expression (43 ⋅ 7^(-2))^-5/7^9 ⋅ 4^(-13) with only positive exponents is (49/43)^5/7^9 ⋅ 4^13.